Problem: Simplify and expand the following expression: $ \dfrac{2}{5y - 9}-\dfrac{y}{y + 2} $
In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(5y - 9)(y + 2)$ Multiply the first term by $\dfrac{y + 2}{y + 2}$ $ \begin{align*} \dfrac{2}{5y - 9} \times \dfrac{y + 2}{y + 2} & = \dfrac{(2)(y + 2)}{(5y - 9)(y + 2)} \\ & = \dfrac{2y + 4}{(5y - 9)(y + 2)}\end{align*} $ Multiply the second term by $\dfrac{5y - 9}{5y - 9}$ $ \begin{align*} \dfrac{y}{y + 2} \times \dfrac{5y - 9}{5y - 9} & = \dfrac{(y)(5y - 9)}{(y + 2)(5y - 9)} \\ & = \dfrac{5y^2 - 9y}{(y + 2)(5y - 9)}\end{align*} $ Now we have: $ = \dfrac{2y + 4}{(5y - 9)(y + 2)} - \dfrac{5y^2 - 9y}{(y + 2)(5y - 9)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{2y + 4 - (5y^2 - 9y)}{(5y - 9)(y + 2)} $ $ = \dfrac{2y + 4 - 5y^2 + 9y}{(5y - 9)(y + 2)} $ $ = \dfrac{11y + 4 - 5y^2}{(5y - 9)(y + 2)}$ Expand the denominator: $ = \dfrac{11y + 4 - 5y^2}{5y^2 + y - 18}$